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The Source of Errors: Thermodynamics

The Source of Errors: Thermodynamics. G A = Activation energy G B = Bond energy. +. 2G B. G B. G A. G A. Correct Growth. Incorrect Growth. Rate of correct growth ¼ exp(-G A ) Probability of incorrect growth ¼ exp(-G A + G B ) Constraint: 2 G B > G A (system goes forward)

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The Source of Errors: Thermodynamics

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  1. The Source of Errors: Thermodynamics GA = Activation energy GB = Bond energy + 2GB GB GA GA Correct Growth Incorrect Growth Rate of correct growth ¼ exp(-GA) Probability of incorrect growth ¼ exp(-GA + GB) Constraint: 2 GB > GA (system goes forward) ) Error probability ¸ exp(-GA/2) ) Rate has quadratic dependence on error probability ) Time to reliably assemble an n £ n square ¼ n5 Ashish Goel, ashishg@stanford.edu

  2. Error-Reducing Designs • Error correction via redundancy: do not change the model • Tile systems are designed to have error correction mechanisms • The Electrical Engineering approach -- error correcting codes • But can not use existing coding/decoding techniques • Proofreading tiles [Winfree, Bekbolatov,’03] • Snake tiles [Chen, Goel ‘04] • Biochemistry techniques • Strand Invasion mechanism [Chen, Cheng, Goel, Huang, Moisset de espanes, ’04] Ashish Goel, ashishg@stanford.edu

  3. Example: Sierpinski Tile System 0 1 0 1 0 0 0 1 0 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  4. Example: Sierpinski Tile System 0 1 0 1 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  5. Example: Sierpinski Tile System 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  6. Example: Sierpinski Tile System 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  7. Growth Error 0 1 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  8. Growth Error 0 1 mismatch 0 1 0 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  9. Growth Error 0 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  10. Growth Error 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  11. Growth Error 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  12. Proofreading Tiles [Winfree, Bekbolatov, ’03] G2 G3 G1 G2a G2b G4 X3 G3b G1b X4 X2 • Each tile in the original system corresponds to four tiles in the new system • The internal glues are unique to this block X1 G3a G1a G4a G4b Ashish Goel, ashishg@stanford.edu

  13. How does this help? 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  14. How does this help? 0 1 mismatch 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  15. How does this help? 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  16. How does this help? No tile can attach at this location 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  17. How does this help? 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  18. How does this help? 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  19. How does this help? 0 1 0 1 0 0 0 1 1 0 1 0 1 1 0 1 Ashish Goel, ashishg@stanford.edu

  20. Nucleation Error Ashish Goel, ashishg@stanford.edu

  21. Nucleation Error First tile attaches with a weak binding strength Ashish Goel, ashishg@stanford.edu

  22. Nucleation Error First tile attaches with a weak binding strength Second tile attaches and secures the first tile Ashish Goel, ashishg@stanford.edu

  23. Nucleation Error First tile attaches with a weak binding strength Second tile attaches and secures the first tile Other tiles can attach and forms a layer of (possibly incorrect) tiles. Ashish Goel, ashishg@stanford.edu

  24. Snake Tiles G2 G3 G1 G2a G2b G4 X2 G3b G1b X1 X3 Each tile in the original system corresponds to four tiles in the new system The internal glues are unique to this block G3a G1a G4a G4b Ashish Goel, ashishg@stanford.edu

  25. How does this help? First tile attaches with a weak binding strength Ashish Goel, ashishg@stanford.edu

  26. How does this help? First tile attaches with a weak binding strength Second tile attaches and secures the first tile Ashish Goel, ashishg@stanford.edu

  27. How does this help? First tile attaches with a weak binding strength Second tile attaches and secures the first tile No Other tiles can attach without another nucleation error Ashish Goel, ashishg@stanford.edu

  28. Preliminary Experimental Results (Obtained by Chen, Goel, Schulman, Winfree) Ashish Goel, ashishg@stanford.edu

  29. Ashish Goel, ashishg@stanford.edu

  30. Ashish Goel, ashishg@stanford.edu

  31. Ashish Goel, ashishg@stanford.edu

  32. Ashish Goel, ashishg@stanford.edu

  33. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  34. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  35. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  36. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  37. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  38. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  39. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  40. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  41. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  42. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  43. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  44. Four by Four Snake Tiles Ashish Goel, ashishg@stanford.edu

  45. Analysis • Snake tile design extends to 2k£2k blocks. • Prevents tile propagation even after k+1 nucleation/growth errors • The error probability changes from p to roughly pk • We can assemble an N£N square in time O(N polylog N) and it remains stable for time W(N) (with high probability). • Resolution loss of O(log N) • Assuming tiles held by strength 3 do not fall off • Matches the time for ideal, irreversible assembly • Compare to N3 for basic proof-reading and N5 with no error-correction in the thermodynamic model [Chen, Goel; DNA ‘04] • Extensions, variations by Reif’s group, Winfree’s group, our group, and others • Recent result: Simple combinatorial criteria; Can avoid resolution loss by using third dimension[Chen, Goel, Luhrs; SODA ‘08] Ashish Goel, ashishg@stanford.edu

  46. Interesting Open Problems - I • General theorems for analyzing reversible self-assembly? • Example: Imagine you are given an “L”, with each arm being length N • From each “convex corner”, a tile can fall off at rate r • At each “concave” corner, a tile can attach at rate f > r • What is the first time that the (N,N) location is occupied? • We believe that the right answer is O(N), can prove O(N log N) • General theorems which relate the combinatorial structure of an error-correction scheme to the error probability? • We have combinatorial criteria for error correction, but they are not all encompassing Ashish Goel, ashishg@stanford.edu

  47. Interesting Open Problems – II Robust, efficient counting • We replace a tile by a k £ k block, where k !1 as N !1 • Or, by a k £ 1 block if we use the third dimension • Codes (eg. Reed-Solomon) can do much better • Can we use codes to design more efficient counters? • Specifically: Do there exist one-to-one functions (code-words) W: {1,..N} ! {1..N2} such that • Given a row of 2 log N tiles encoding W(k), there is some simple “tiling subroutine” to assemble W(k+1) on top • Even if there are p log N errors in the tiling process for each row, this process stops after “counting” from 1 to N • Motivation: Correctly assembling large shapes up-to molecular precision will be a new engineering paradigm – so an exciting opportunity for theoreticians Ashish Goel, ashishg@stanford.edu

  48. 1W (1,1) (0,1) (1,1) (0,0) (0,0) (0,0) 1W (1,0) (1,0) (1,0) (0,0) (0,0) (0,0) 1W (1,1) (0,0) (0,0) (0,1) (1,1) (0,0) 1W (1,1) (0,0) (0,1) (1,0) (1,1) 1W (0,1) (1,1) (0,1) (1,1) (0,1) 1S 1S 1S 1S 1S 1S Another Mode of Error -- Damage (0,1) S (1,0) (1,1) 1W (0,0) (1,0) (0,0) (0,1) 1S (1,0) (1,1) (1,1) S Ashish Goel, ashishg@stanford.edu

  49. What went wrong? • When tiles attach from unexpected directions the “correct” tile is not guaranteed. • Potential fix: Design systems more carefully so that the system can reassemble from small pieces all over. • Previous work: [Winfree ’06] Rectilinear Systems that will grow back correctly as long as the seed remains in place by forcing growth only from the seed direction. • Single point of failure: Lose the seed and the structure cannot regrow • Akin to a lizard regenerating a limb • Our goal: Tile systems that heal from small fragments anywhere • Akin to two parts of a starfish growing into complete separate starfish • Almost a “reproductive property” Ashish Goel, ashishg@stanford.edu

  50. Two pieces of self-healing: Immutability and Progressiveness Immutability: Only correct tiles may attach. (As opposed to the Sierpinski example.) Progressiveness: Eventually, all tiles attach. (Provided one of a set of pieces containing enough information remains) Example: The Chinese remainder counter is almost self-healing from any row Ashish Goel, ashishg@stanford.edu

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